The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices

This paper proposes scalable and fast algorithms for solving the Robust PCA problem, namely recovering a low-rank matrix with an unknown fraction of its entries being arbitrarily corrupted. This problem arises in many applications, such as image processing, web data ranking, and bioinformatic data analysis. It was recently shown that under surprisingly broad conditions, the Robust PCA problem can be exactly solved via convex optimization that minimizes a combination of the nuclear norm and the $\ell^1$-norm . In this paper, we apply the method of augmented Lagrange multipliers (ALM) to solve this convex program. As the objective function is non-smooth, we show how to extend the classical analysis of ALM to such new objective functions and prove the optimality of the proposed algorithms and characterize their convergence rate. Empirically, the proposed new algorithms can be more than five times faster than the previous state-of-the-art algorithms for Robust PCA, such as the accelerated proximal gradient (APG) algorithm. Moreover, the new algorithms achieve higher precision, yet being less storage/memory demanding. We also show that the ALM technique can be used to solve the (related but somewhat simpler) matrix completion problem and obtain rather promising results too. We further prove the necessary and sufficient condition for the inexact ALM to converge globally. Matlab code of all algorithms discussed are available at http://perception.csl.illinois.edu/matrix-rank/home.htmlComment: Please cite "Zhouchen Lin, Risheng Liu, and Zhixun Su, Linearized Alternating Direction Method with Adaptive Penalty for Low Rank Representation, NIPS 2011." (available at arXiv:1109.0367) instead for a more general method called Linearized Alternating Direction Method This manuscript first appeared as University of Illinois at Urbana-Champaign technical report #UILU-ENG-09-2215 in October 2009 Zhouchen Lin, Risheng Liu, and Zhixun Su, Linearized Alternating Direction Method with Adaptive Penalty for Low Rank Representation, NIPS 2011. (available at http://arxiv.org/abs/1109.0367

Structural Mechanism of Trimeric HIV-1 Envelope Glycoprotein Activation

<div><p>HIV-1 infection begins with the binding of trimeric viral envelope glycoproteins (Env) to CD4 and a co-receptor on target T-cells. Understanding how these ligands influence the structure of Env is of fundamental interest for HIV vaccine development. Using cryo-electron microscopy, we describe the contrasting structural outcomes of trimeric Env binding to soluble CD4, to the broadly neutralizing, CD4-binding site antibodies VRC01, VRC03 and b12, or to the monoclonal antibody 17b, a co-receptor mimic. Binding of trimeric HIV-1 BaL Env to either soluble CD4 or 17b alone, is sufficient to trigger formation of the open quaternary conformation of Env. In contrast, VRC01 locks Env in the closed state, while b12 binding requires a partial opening in the quaternary structure of trimeric Env. Our results show that, despite general similarities in regions of the HIV-1 gp120 polypeptide that contact CD4, VRC01, VRC03 and b12, there are important differences in quaternary structures of the complexes these ligands form on native trimeric Env, and potentially explain differences in the neutralizing breadth and potency of antibodies with similar specificities. From cryo-electron microscopic analysis at ∼9 Å resolution of a cleaved, soluble version of trimeric Env, we show that a structural signature of the open Env conformation is a three-helix motif composed of α-helical segments derived from highly conserved, non-glycosylated N-terminal regions of the gp41 trimer. The three N-terminal gp41 helices in this novel, activated Env conformation are held apart by their interactions with the rest of Env, and are less compactly packed than in the post-fusion, six-helix bundle state. These findings suggest a new structural template for designing immunogens that can elicit antibodies targeting HIV at a vulnerable, pre-entry stage.</p> </div

Structure of the open conformation of trimeric Env at sub-nanometer resolution.

<p>(a) Side view of the structure of trimeric Env bound to 17b Fab. The map was fitted with three copies of the X-ray structures for the gp120-17b portion of the 1GC1 coordinates with gp120 (red) and 17b Fv fragments (light chain: yellow, heavy chain: green). One copy of the gp41 N-terminal helix (cyan) of 1AIK coordinates (N34) was fitted individually into each of the three densities, which occupy the central region of the spike that is essentially a cavity in the unliganded state. (b) Side view of the density map from unliganded native trimeric Env <a href="http://www.plospathogens.org/article/info:doi/10.1371/journal.ppat.1002797#ppat.1002797-Liu1" target="_blank">[21]</a>, <a href="http://www.plospathogens.org/article/info:doi/10.1371/journal.ppat.1002797#ppat.1002797-White1" target="_blank">[22]</a>, <a href="http://www.plospathogens.org/article/info:doi/10.1371/journal.ppat.1002797#ppat.1002797-Harris1" target="_blank">[23]</a>, <a href="http://www.plospathogens.org/article/info:doi/10.1371/journal.ppat.1002797#ppat.1002797-White2" target="_blank">[24]</a>, with the three gp41 N-terminal helices (cyan) superposed to show that in the open conformation, they occupy the solvent filled cavity in the density map of the unliganded state.</p